An Overview on Robust Control

Transcription

1 Advanced Control An Overview on Robust Control P C Scope Keywords Prerequisites allow the student to assess the potential of different methods in robust control without entering deep into theory. Sensitize for the necessity of robust feedback control. uncertainty representations, H, µ synthesis, LMI Nyquist criterion, gain and phase margin, LQG state space control Contact Raoul Herzog 1, Jürg Keller 2 Version 5.1 Date September 19, raoul.herzog@heig vd.ch 2 juerg.keller1@fhnw.ch

2 Advanced Control, An Overview on Robust Control MSE 2 Raoul Herzog, Jürg Keller September 19, 211

3 Contents 1 Introduction to Robust Control Motivation of Robust Control An Attempt to define Robust Control Structure of this Document Review of Norms for Signals and Systems The Nyquist Criterion and the Small Gain Theorem Review of the Nyquist Criterion and Classical Stability Margins The Small Gain Theorem Applications of the Small Gain Theorem to Robust Control Description of Model Uncertainty Unstructured Uncertainty Structured Uncertainty Formulation of the Standard H Problem A Glimpse on the H State Space Solution Limitation of H Methods Outlook: µ Synthesis and LMI Methods Structured Singular Values (SSV) and µ Synthesis Linear Matrix Inequalities (LMI) Conclusion A.1 Examples A.1.1 Example A.1.2 Example A.2 Exercise 1: Linear Fractional Transformation A.3 Exercise 2: Small gain theorem A.4 Exercise 3: Drawback of classical stability margins A.5 Exercise 4: Two cart problem Bibliography September 19, 211 Raoul Herzog, Jürg Keller 3

4 Advanced Control, An Overview on Robust Control MSE 1 Introduction to Robust Control 1.1 Motivation of Robust Control System modeling is generally one of the most important tasks in engineering, and in particular in control engineering. Modeling is always a delicate task, since the physical reality is often complicated, and all attempt to mathematically describe a real physical process involves simplifications and assumptions, e.g.: dynamics are often consciously neglected 1 to make the model tractable, e.g. unmodeled sensor and actuator dynamics, higher order modes from large scale structures modeled by finite elements (FEM), nonlinearities are often either hard to model or too complicated, parameters are often not exactly known, either because they are hard to measure precisely, or because of varying manufacturing conditions 2. Furthermore, it is desirable to end up with simple plant models. Indeed, in modern control, the controller is the output of an optimization problem, and the complexity of the controller is directly linked with the complexity of the plant model: a complicated high order plant will automatically lead to a complicated high order controller, which is undesirable. Robust control deals with system analysis and control design for such imperfectly known process models. One of the main goals of feedback control is to maintain overall stability and system performance despite uncertainties in the plant. In general, robustness does not come for free from a controller designed via optimal control and estimation theory (observer design): a controller designed for a nominal process model generally works fine for the nominal plant model, but may fail 3 for even a nearby plant model. An important point of all feedback control synthesis methods is the control engineer s awareness of inherent trade offs : increasing the robustness will generally make the controller less aggressive, and will thereby decrease system performance. Robust control allows to specify more or less directly the plant uncertainty, and allows to predict the possible trade offs between robustness and closed loop performance. Sometimes bachelor level courses in control may give the impression that everything is feasible with control, and that it s only a matter of finding a good controller. But this impression is completely false: the plant itself implies inherent limitations 4, especially if the plant is unstable. The achievable performance/robustness 1 It s common to say: My model is precise up to... Hz. 2 For serial production of devices incorporating feedback control, individual tuning is not desirable. The robustness of the system should be sufficient to tolerate all uncertainties and tolerances from the manufacturing processes. For mechatronical systems, the manufacturing uncertainties are coming from the mechanical and electronical parts. Generally, there are no manufacturing tolerances inside the digital controller because software is perfectly reproducable. 3 Instability or unacceptable performance degradations may occur. 4 e.g. by the Bode integral relation. 4 Raoul Herzog, Jürg Keller September 19, 211

5 MSE Advanced Control, An Overview on Robust Control trade offs strictly depend on the plant, and cannot be overcome by any sophisticated control [1]. 1.2 An Attempt to define Robust Control A definition of robust control could be stated as: Robust control aims at designing a fixed (non adaptive) controller such that some defined level of performance 5 of the controlled system is guaranteed, irrespective of changes in plant dynamics within a predefined class. Robust control offers a collection of powerful mathematical tools and efficient software algorithms able to partly 6 answer the following key questions: Describing and characterizing plant uncertainty: What is a good way to describe plant variations or plant uncertainty? Some descriptions attempt to faithfully describe the real situation (e.g. probability distributions on physical parameters), but unfortunately there may exist no efficient method to solve the problem. Other uncertainty descriptions are less direct, but more convenient for the theory (e.g. analytic solutions exist). Robustness analysis problems: As an example consider a nominal plant and a given stabilizing controller. The uncertainty in the plant model is defined by a class of perturbations. There is no true model, we have to deal with a given set of possible plant models. Now, we would like to know whether or not the closed loop remains stable for the whole class of plants. This is a typical robustness analysis problem. Robustness synthesis problems: Find a controller which stabilizes a given class of plants. Generally, synthesis problems are more difficult to solve than analysis problems. 1.3 Structure of this Document The document is structured as follows: In section 2, norms for signals and systems are reviewed for the single input single output case (SISO). Section 3 recapitulates the Nyquist criterion and the small gain theorem, an important working horse in robust control. Section 4 addresses the question how to describe model uncertainty. Sections 5 introduces the setup of H control. It is not the goal of this document to describe the underlying mathematical theory, which is quite demanding. Therefore, 5 e.g. closed loop stability, reference tracking performance, and disturbance rejection performance. 6 It should be noted that some simple problems in robust control (e.g. exact stability determination of a linear system in which several parameters vary over given ranges) have shown to be NP hard, hence as difficult as other famous problems for which no efficient solutions are known to exist, or likely to be found. September 19, 211 Raoul Herzog, Jürg Keller 5

6 Advanced Control, An Overview on Robust Control MSE section 6 only sketches the H solution from a user point of view. Section 7 discusses some limitations and drawbacks of standard H methods. Finally, section 8 gives an outlook to the actual state of the art in robust control. Section 9 concludes with some general remarks on robust control. 2 Review of Norms for Signals and Systems Norms are used in many places of engineering to quantify the magnitude of an object (e.g. the amplitude of a signal), or to quantify the proximity of two objects (e.g. the proximity of two systems). In robust control, norms play a crucial role: the choice of metric used to quantify the amount of process uncertainty, the choice of norm used in the optimization problem associated with the controller synthesis. The linear quadratic gaussian LQG control is based on the optimization of a 2 norm, whereas H control is based on the optimization of a norm. The L 2 (Euclidean) norm of a time domain scalar signal u(t) is defined as: u 2 = u 2 (t)dt. (1) If this integral is finite, then the signal u is square integrable, denoted as u L 2. For vector valued signals u(t), u(t) = u 1 (t) u 2 (t). u n (t). (2) the 2 norm is defined as u 2 = u(t) 2 2 dt = u T (t)u(t) dt, (3) where the superscript T denotes transposition. A Linear Time Invariant (LTI) system G can be described either by a state space realization or by its corresponding transfer function ẋ = Ax + Bu y = Cx + Du G(s) = C(sI A) 1 B + D. (4) 6 Raoul Herzog, Jürg Keller September 19, 211

7 MSE Advanced Control, An Overview on Robust Control Similar to signals, we would like to define different norms for systems, respectively for transfer functions. Two systems G 1 and G 2 are close if the norm of the difference of their transfer functions G 1 G 2 is small. Two systems might be close with respect to one defined norm, but far with respect to another norm 7. The H 2 norm of a stable system G(s) is defined as the L 2 norm of its impulse response g(t). When dealing with stochastic signals, the H 2 norm of a system corresponds to the variance E[ y 2 ] of the system response y(t) when the system is excited by a unit intensity white noise input u(t). The H norm of a stable transfer function G(s) is defined as the maximum RMS amplification over arbitrary square integrable input signals u. G = sup u L 2 y 2 u 2 (5) Note that the H norm for a system is induced by the L 2 norm for signals. The physical interpretation of the H norm corresponds simply to the maximum energy amplification over all input signals. It can be shown that for single input single output systems G equals the peak magnitude in the Bode diagram of the transfer function G(j ω): G = sup G(j ω) (6) ω In H control the performance to be optimized is defined in terms of minimizing the H norm of closed loop transfer functions, e.g. the sensitivity function S(s) and the complementary sensitivity function T(s). This type of optimization problem is also called min max problem: H control seeks to minimize the worst case scenario, i.e. when the closed loop function has its peak. In contrast, classical LQG control minimizes the closed loop behaviour for known input signals 8, whereas H control works with unknown input signals and tries to optimize the worst case scenario. Therefore, the solution of H control problems typically inhibits flat 9 Bode magnitude plots (no more peak). When minimizing the absolute peak in the Bode magnitude diagram, automatically other local peaks pop up. This effect is called waterbed effect. Finally, the value of all local peaks join the value of the absolute peak, and the response becomes flat. The theoretical background of this comes from the Bode integral theorem, see equation (12) and figure 6 page 12. Instead of flat Bode magnitude plots we can also impose a given shape. This corresponds to a modern version of classical loop shaping. In this context, important design parameters are frequency dependent weighting functions W(s). They allow 7 With the metric induced by the H norm, the two systems P 1 (s) = 1 s+ǫ and P 2(s) = 1 s are infinitely far, no matter how small ǫ is. This may seem illogical because a reasonable controller will stabilize both plants P 1 and P 2 simultaneously, and the resulting closed loop systems will be nearly identical. A remedy consists in using the so called Vinnicombe metric, which also allows to treat marginally stable and unstable systems. 8 e.g. the energy of the resulting closed loop impulse response 9 allpass behaviour September 19, 211 Raoul Herzog, Jürg Keller 7

8 Advanced Control, An Overview on Robust Control MSE 1 2 Bode Diagram Magnitude (abs) Phase (deg) Frequency (Hz) Figure 1: Example: The norm of G(s) = 1 s 2 +.1s+1 is G 1. to shape given closed loop functions, e.g. the sensitivity S(s), by optimizing their weighted norm W S. The H norm of a multivariable (MIMO) transfer matrix G(s) needs the important concept of singular values which is beyond the scope of this document. 3 The Nyquist Criterion and the Small Gain Theorem The Nyquist criterion is a cornerstone of classical control, and is of fundamental importance in robust control. The following chapter recapitulates the Nyquist criterion, the small gain theorem, and shows its application to robust control. 3.1 Review of the Nyquist Criterion and Classical Stability Margins The Nyquist criterion allows to check closed loop stability based on the inspection of the loop gain L(s) = C(s) P(s), without computing the closed loop poles, i.e. the roots of 1 + L(s) =. The Nyquist criterion is based on Cauchy s argument, and says: The closed loop system with loop gain L(s) and a negative feedback polarity is stable if and only if the complete 1 Nyquist plot of L(jω) encircles the critical point s crit = 1 exactly N P anticlockwise times in the complex plane, where N P is the number of unstable (right halfplane) poles of L(s). As a special case, if L(s) already is a stable open loop transfer function, N P is zero, and the Nyquist plot of L(jω) must not encircle the critical point 1 in order 1 For the complete Nyquist plot, ω runs from to +. 8 Raoul Herzog, Jürg Keller September 19, 211

9 MSE Advanced Control, An Overview on Robust Control 1 1 Waterbed effect minimize the peak! magnitude will pop up elsewhere! magnitude of a closed loop function frequency Hz Figure 2: Illustrating the waterbed effect. loop gain L(s) feedback with minus polarity Figure 3: Elementary feedback system with loop gain L(s) to ensure closed loop stability. Figure 4 recapitulates the definition of the classical gain margin A m and phase margin φ m. For example a gain margin of A m = 2 means that the closed-loop stays stable even if the loop gain doubles. The phase margin φ m indicates the amount of additional delay the feedback loop can tolerate before becoming unstable. Robust control does not work with the classical stability margins A m and φ m for two reasons: 1) there exist no analytical optimization techniques for these margins, and 2) there are cases where the classical margins indicate a good robustness against individual gain and phase tolerances, whereas the feedback loop is not at all robust against simultaneous variations of gain and phase, see exercice 2, page 48. For these reasons, robust control prefers as margin the critical distance d crit between the Nyquist plot of L(jω) and the critical point s crit = 1: d crit = min ω (dist(l(jω),s crit )) = min ω L(jω) s crit = min ω 1 + L(jω). (7) It follows that the critical distance d crit is the reciprocal of the sensitivity function September 19, 211 Raoul Herzog, Jürg Keller 9

10 Advanced Control, An Overview on Robust Control MSE Nyquist plot of L(s) complex plane 1.5 imaginary part 1 d crit 1/A m L(j ) L(j ).5 φ m 1 L(j ω) real part Figure 4: Definition of the critical distance d crit and the classical stability margins. Here, the Nyquist plot is only drawn for positive frequencies ω >. peak: d crit = 1 S, where S(jω) = L(jω). (8) Maximizing the critical distance d crit corresponds to minimizing the norm of the sensitivity function. Therefore, one of the natural objectives 11 of robust control consists of minimizing the norm of the sensitivity function. 3.2 The Small Gain Theorem The small gain theorem follows directly from the Nyquist criterion: if L(s) is stable and L < 1, then the loop gain L(jω) is smaller than 1 for all frequencies ω, and hence L(jω) does not encircle the critical point 1. It follows that the small gain condition L < 1 is a sufficient but not necessary condition for closed loop 11 It is important to see that sensitivity peak minimization is not the only objective in real world problems. In fact, minimizing only the norm of the sensitivity function turns out to be an ill-posed problem, because the gain of the resulting controller would be infinite. 1 Raoul Herzog, Jürg Keller September 19, 211

11 MSE Advanced Control, An Overview on Robust Control stability. The small gain condition corresponds to an infinite phase margin φ m =. If L < 1, the closed loop stays stable even if the feedback polarity is wrong! The small gain theorem is not limited to linear feedback control: it can even be generalized to nonlinear feedback control. In this case, L becomes a nonlinear operator in time domain, and the norm condition on L(s) must be replaced by an operator norm condition. This may sound abstract, but a simple application is the feedback of a linear dynamic system in cascade with a static nonlinearity. In this case, the small gain theorem yields a sector bounded 12 nonlinearity as a sufficient condition for closed loop stability. It makes no sense to apply the small gain theorem directly for controller synthesis: for good tracking performance a high loop gain is needed within the control bandwidth 13. However, the small gain theorem has a great utility for analyzing feedback loops with unstructured uncertainty. 3.3 Applications of the Small Gain Theorem to Robust Control Suppose that P (s) denotes the nominal plant, and C(s) a stabilizing controller. Now consider an additive perturbation which yields a cloud of plants P(s) = P (s)+ a (s), where a (s) denotes an unknown stable transfer function representing the modeling uncertainty of P. The question arises how much uncertainty the closed loop may tolerate before becoming unstable. Figure 5 shows that the feedback perturbed plant P additive uncertainty a nominal plant + a P -C + + feedback seen by P + controller C a -C 1+P C Figure 5: Application of the Small Gain Theorem to additive plant uncertainty. 12 also called Popov criterion. 13 If the controller includes an integral action (e.g. the I part of PID), the loop gain is infinite at Hz, and hence L =. September 19, 211 Raoul Herzog, Jürg Keller 11

12 Advanced Control, An Overview on Robust Control MSE C seen by a is. The corresponding loop gain is C 1+P C a. Since both 1+P C a and the nominal closed loop are stable we can apply the small gain theorem for the feedback loop seen by a. A sufficient condition for the stability of the perturbed system is therefore: C a < 1. (9) 1 + P C Using the submultiplicative property of norms AB A B, the sufficient stability condition becomes: C 1 a < 1 = a 1 + P C < C. (1) 1+P C Equation (1) is a conservative bound indicating the amount of tolerated additive uncertainty which preserves stability. In case of multiplicative unstructured uncertainty, the set of perturbed plants is described as P(s) = P (s) (1+ m (s)). Using the small gain theorem, the following stability bound can be found: m < 1 P C = 1+P C 1 T. (11) A high complementary sensitivity peak value T leads to a small tolerable multiplicative perturbation m. We will now discuss an extremely fundamental relationship, called Bode integral theorem: ln S(jω) dω = π Re(p). (12) unstable poles p of L(s) This law can be seen as a conservation law: the integrated value of the log of the magnitude of sensitivity function S(jω) is conserved under the action of feedback. If the open loop L(s) is stable, then the integral becomes zero. At low frequencies, in order to have good tracking performance, the sensitivity must be much smaller than 1 (negative db levels), i.e. ln S(jω) must become negative. The Bode integral theorem states that the average sensitivity improvement at low frequencies is compensated by the average sensitivity deterioration at high frequencies. This is illustrated by figure 6. If the plant is unstable, the situation is becoming worse since the right hand side of equation (12) is now positive. This means that the average sensitivity deterioration is always larger than the improvement. The more unstable the plant is, the more positive the real part of the unstable poles, the more difficult the situation becomes. This applies to every controller, no matter how it was designed. Unstable plants are inherently more difficult to control than stable plants [1]. 4 Description of Model Uncertainty It is worthwhile to clarify what is meant by model uncertainty. In a control system there are two categories of uncertainties: disturbance signals and perturbations in 12 Raoul Herzog, Jürg Keller September 19, 211

13 MSE Advanced Control, An Overview on Robust Control 1 Log Magnitude Frequency 2. Figure 6: Loopshaping constraints: Sensitivity reduction at low frequencies inevitably leads to sensitivity increase at higher frequencies. Picture taken from [1]. the plant dynamics. Disturbances are external stochastic inputs which are not under control. Examples of disturbance signals are sensor and actuator noise or changes of the environment. Dynamic perturbations are model uncertainties caused by not exactly known or slowly changing plant parameters, and unmodeled or approximated system dynamics. In a house temperature control, disturbances and dynamic plant perturbations would be : Disturbances: changing outdoor temperature, wind speed, open windows and doors, heating due to electrical equipment or human bodies, sun radiation Dynamic plant perturbations: not exactly known isolation coefficients, unknown and changing heat capacities, uncertain efficiency of the heating device The example shows that it is not always clear how to classify the disturbances. Therefore, it is not surprising that in µ synthesis the two categories of disturbances are handled within the same formalism. For H controller design plant uncertainty results from approximating the real system with a mathematical model of tractable complexity. Additionally its physical parameters are not exactly known. As explained in the motivation section, the aim of H based controller design is to include uncertainty into controller design. To achieve this it is necessary to have a mathematical description of model uncertainty. In this section, two different types of uncertainty descriptions will be introduced. The first is unstructured uncertainty, whereas the second is structured uncertainty. 4.1 Unstructured Uncertainty A description of model uncertainty has to meet the following goals: it should be simpler than a physical model of the neglected system dynamics, and it should be September 19, 211 Raoul Herzog, Jürg Keller 13

14 Advanced Control, An Overview on Robust Control MSE tractable with a formalism which can easily be used in controller design. A simple solution is found, if all dynamics, time invariant perturbations that may occur in different parts of the system, are represented with a single perturbation transfer function (s). There are many possibilities to include (s) into a control system, but only the two most commonly used will be presented in the following. These are shown in figures 7 and 8. The focus is on SISO systems. Figure 7 shows a (s) as an additive perturbation of the plant transfer function P(s): P(s) = P o (s) + a (s), (13) where P(s) is the actual, perturbed plant, and P (s) is the nominal plant. The transfer function a (s) is used to describe a frequency dependent unstructured uncertainty as follows: a (s) = W 2 (s) a (s) (14) where the normalized perturbation a (s) is any stable transfer function with: a 1 (15) This uncertainity description is closely related to a plant transfer function representation in C, the Nyquist plot, as shown in figure 9a). a = 1 defines a circle, whose radius is scaled 14 with W 2 (s). a Figure 7: Additive uncertainty Dynamic perturbations can also be described with multiplicative uncertainty. The corresponding block diagram is shown in figure 8. P(s) = P (s) (1 + m (s)) (16) In multivariable systems, transfer functions are matrices. It is known that matrix multiplication is not commutative, so for matrix valued P and m, P (s) (I + m (s)) is generally different from (I + m (s)) P (s). As a consequence, input and output multiplicative perturbations have to be distinguished. Input multiplicative perturbations are able to model actuator uncertainty, whereas output multiplicative perturbations are used to describe sensor related uncertainties. 14 Index 2 for W is commonly used for uncertainty weighting functions. 14 Raoul Herzog, Jürg Keller September 19, 211

15 MSE Advanced Control, An Overview on Robust Control m Figure 8: Multiplicative uncertainty at plant input The capabilities of additive or multiplicative uncertainty representation are illustrated with an example. For comparison purposes the unmodeled high frequency dynamic is specified. It will be investigated how this a priori known model error can be represented with an additive or multiplicative uncertainty. Example: Many plants can roughly be approximated by a first order system, e.g. PT1. Such a model usually neglects the high frequency dynamic of the system. As it is well known, a control system with a PT1-plant can achieve any required closed loop performance. Consequently, we are interested in an uncertainty description which incorporates possible unmodeled phase loss into controller design and prevents a controller design with unrealistic high bandwidth. The nominal plant model is P (s) = 5 s + 1. (17) The unmodeled dynamic is represented with a set of transfer functions with the following elements: P (s) = The plants to be controlled are: 1 (τs + 1) 2, τ [.2...5]. (18) P(s) = P o (s) P (s). (19) For comparison purposes the unmodeled dynamic described above is represented as an additive and multiplicative perturbation. Additive uncertainties are best represented in the complex plane C, i.e. in the Nyquist diagram of figure (9a), whereas multiplicative uncertainties can easily be plotted in a Bode diagram of figure (9b). The figures show the plant s frequency response P(jω) for different values of τ. Next, a single description of uncertainty has to be found for the specified range of values for the uncertain parameter τ. To accomplish this, the additive and multiplicative (jω) are determined for a set of values of τ according to the following formulas: September 19, 211 Raoul Herzog, Jürg Keller 15

16 Advanced Control, An Overview on Robust Control MSE Additive unstructured perturbation: a (s) = P (s) P (s). (2) Multiplicative unstructured perturbation: m (s) = P (s) 1. P (s) (21).5 nominal plant perturbed plant set (a) Additive uncertainty (b) Multiplicative uncertainty Figure 9: Plant uncertainty The magnitude of the perturbations a (jω) and m (jω) are shown in figure 1. As defined in (14), frequency dependency is specified with the weighting transfer function W2 (s). The solid line in the figures is an envelope of all error frequency responses and (s) = W2 (s) (s) is therefore an upper bound for the uncertainty. Additive weighting: W2a (s) = 7.5s. (s + 1)(s + 15) (22) W2m (s) =.997s(s + 17). (s + 9)(s + 135) (23) Multiplicative weighting: In figure 11 the resulting uncertainty regions are plotted in the corresponding plots. Compare the resulting regions with the set of plants in figure 9! Obviously, the uncertainty region is very large due to the fact, that the nominal plant is not in the center of the plant set. Since the unnecessary uncertainty regions are mainly opposite of the critical point, it might not have a large impact on controller design. After ω = 1 rad/s the gain point lies within the uncertainty set. As a consequence, a 16 Raoul Herzog, Ju rg Keller September 19, 211

17 MSE Advanced Control, An Overview on Robust Control 1 approx. error bound additive. errors 1 1 approx. error bound multipl. errors (a) Additive uncertainty (b) Multiplicative uncertainty Figure 1: Uncertainty descriptions.5.5 perturbed plant set(w) (a) Additive bounds (b) Multiplicative bounds Figure 11: Uncertainty bounds plant phase cannot be determined anymore. In the Bode diagram of figure phase bounds are set to 27 and 9 degrees, in order to get a nice bode plot. In a similar way, static nonlinearities (sector criterion) or uncertain dead time can be described with unstructured uncertainties. There arises the question, which of the representations of uncertainty should be used. Since the optimal H controller has the order of the plant plus the orders of all the weighting functions, it is preferable to choose a representation which leads to a minimal order weighting. In the SISO case, additive uncertainty can be recast into multiplicative uncertainty with simple algebraic operations: P (s) m (s) = a (s) (24) Since the additive representation includes the plant transfer function P (s) it is expected, that an additive representation usually is of higher degree. To illustrate this: uncertain dead time can be fit into a multiplicative uncertainty description September 19, 211 Raoul Herzog, Jürg Keller 17

18 Advanced Control, An Overview on Robust Control MSE which is independent of the the plant: P(s) = P o (s) e st (25) m (s) = e st 1 (26) This function is shown in the amplitude plot of figure 12. It can be bounded with a DT1 transfer function Figure 12: Dead time uncertainty The chosen uncertainty representation determines which closed loop transfer function has to be considered in H controller design in order to guarantee robustness with respect to stability and performance. Additive and multiplicative uncertainty are special cases of a more general framework which will be explained below. First, notice that we have to distinguish two different feedback loops: the feedback loop formed by the controller and the perturbed plant, and the feedback loop in which the uncertainty resides. A general solution for all different uncertainty descriptions can be obtained, if the control system is described in the feedback structure shown in figure 13. The plant P is partitioned according to the dimensions of. P = [ P11 P 12 P 21 P 22 With simple algebraic calculations it follows: ]. (27) z = [P 22 + P 21 (I P 11 ) 1 P 12 ] w (28) 18 Raoul Herzog, Jürg Keller September 19, 211

19 MSE Advanced Control, An Overview on Robust Control h g Figure 13: Standard P configuration if (I P 11 ) 1 exists. Define: F(P, ) := P 22 + P 21 (I P 11 ) 1 P 12 (29) F(P, ) is called a linear fractional transformation LFT of P and. In the single input single output case the LFT corresponds to a bilinear transform. This representation is also used for more sophisticated uncertainty representation (next section) and also to formulate the general H controller design problem. Additive and multiplicative uncertainty representations are special cases of a linear fractional transformation. For additive perturbations, the matrix P becomes: [ ] I P =. (3) I P For multiplicative perturbations at the plant output, the matrix P becomes: [ ] P P =. (31) I P The example before shows that the conservative error bounds on the plant can be large, compared to the real plant perturbation. An unstructured uncertainty model is also not suited for perturbations which only affects a part of an interconnected system. This motivates a more general representation of uncertainty which will be introduced in the next section. 4.2 Structured Uncertainty Structured uncertainty means uncertainty (tolerances) of concrete physical parameters. Some examples are listed below: electrical components like resistors or capacitors are always affected with tolerances, e.g. 2% for a standard resistor. In a magnetic bearing system, the nominal air gap is an important parameter which is affected by manufacturing tolerances and by thermal growth when the machine is running. September 19, 211 Raoul Herzog, Jürg Keller 19

20 Advanced Control, An Overview on Robust Control MSE The relative permeability µ r of a magnetic material is usually not very precisely known The internal structure of a linear plant can be represented as a block diagram with integrators, summators, and gains. The physical parameters are gains in the block diagram. It can be shown that the plant transfer function is always an LFT (linear fractional transform) with respect to every physical parameter p 1,...,p n. As an example, consider the uncertain plant equation (17), (18) and (19) page 15. We will treat the time constant τ in P (s) as uncertain parameter lying in the range of Suppose that the nominal value τ corresponds to the mid range value τ =.35, and τ = τ + τ, where τ is a real parameter varying between The uncertain system P(s) = P (s) P (s) = 5 1 can s+1 (τs+1) 2 be recasted with the following linear fractional transform in figure 14. Note that diagonal uncertainty block h g w P aug z Figure 14: Structured uncertainty with an unknown time constant τ appearing twice. the uncertainty block in figure 14 is diagonal with the same diagonal element τ repeated twice. The reason is that the uncertain time constant τ appears twice in the second order term P (s). The augmented plant P aug (s) will be of third order with 3 inputs and 3 outputs. Exercise : Calculate P aug (s) and show that the associated linear fractional transform yields F(P aug, ) = 5 s ((τ + τ )s + 1) 2 (32) There are available software tools capable to directly define and handle systems with structured or unstructured uncertainty. Using the Robust Control toolbox of Matlab the example above can be entered as follows : >> P = tf(5, [1, 1]) Transfer function: 2 Raoul Herzog, Jürg Keller September 19, 211

21 MSE Advanced Control, An Overview on Robust Control s + 1 >> tau = ureal( tau,.35, range, [.2,.5]) Uncertain Real Parameter: Name tau, NominalValue.35, Range [.2.5] >> Pdelta = tf(1, [tau, 1])*tf(1, [tau, 1]) USS: 2 States, 1 Output, 1 Input, Continuous System tau: real, nominal =.35, range = [.2.5], 2 occurrences >> P = P * Pdelta USS: 3 States, 1 Output, 1 Input, Continuous System tau: real, nominal =.35, range = [.2.5], 2 occurrences >> bode(p) which gives the family of Bode plot in figure Bode Diagram Magnitude (db) Phase (deg) Frequency (rad/sec) Figure 15: Example : structured uncertainty with an unknown time constant τ. In contrast to unstructured uncertainty, see figure 13, structured uncertainty leads to a feedback configuration figure 15 with a static diagonal block. In addition, the diagonal terms are real valued, whereas in the unstructured case they are mostly complex valued norm bounded transfer functions (s). It is obvious that structured uncertainty representations are often closer to physical specifications. However, the algorithms needed to tackle robust control analysis or synthesis problems with structured uncertainty are much more complicated, see outlook section page 27. September 19, 211 Raoul Herzog, Jürg Keller 21

22 Advanced Control, An Overview on Robust Control MSE We disadvise from recasting highly structured parameter uncertainty as additive or multiplicative unstructured uncertainty, because this implies a high degree of conservativeness. Indeed, the class of perturbations may become excessively large, and the resulting controller, if there is any complying with the robustness specifications is likely to show poor performance. 5 Formulation of the Standard H Problem In this section the controller design problem is formulated in the H formalism. The objective is to design a controller, which does not only have nominal stability and performance, but has this properties with all the plants represented by some uncertainty description. H controller design was specially developped, to solve this problem in a systematic manner. As system which is stable with all the plants of the uncertainty set is called robustly stable. It has a robust performance, if the performance specifications are met for all the admissible plant behaviors. The control system with controller and uncertainty model is shown in figure 16. In H controller design, the H norm of the mapping from w to z can be minimized. As a consequence the control problem has to be formulated in this formalism. The inputs w are typically reference or disturbance signals, whereas the outputs z can be the control error or the controller output. The H controller design problem Figure 16: Control system with uncertainty cannot directly by solved in the structure of figure 16. Since performance specifications and stability conditions can both be expressed as H norm conditions on some transfer functions, there are two ways to simplify the control system of figure 16. Either the stability conditions are formulated in the same formalism as performance specifications, resulting in a structure as in figure 17(a), or the performance specification is formulated similarly to the uncertainty description p (see figure 17(b)). For unstructured uncertainties the structure of figure 17(a) is easier to go, but for structured uncertainties, only the structure in figure 17(b) leads to an elegant representation. The structured uncertainty problem is solved with µ-synthesis, which is beyond the scope of this introduction. Consequently, the first approach will be followed in the sequel. As a result from the small gain section, it is clear that a system is robustly stable, if 22 Raoul Herzog, Jürg Keller September 19, 211

23 MSE Advanced Control, An Overview on Robust Control (a) Robust Stability as norm condition (b) Performance as artificial plant uncertainty Figure 17: Possible system representations Figure 18: Performance Weighting Function for additive uncertainty: for multiplicative uncertainty: W 2 C(I + PC) 1 1 (33) W 2 PC(I + PC) 1 1 (34) In a control system as shown in figure 19 the robust stability condition corresponds to the H norm of the mapping from r to u for the additive case and from r to y for the multiplicative case. System performance is usually specified by conditions on disturbance attenuation, i.e. on the H norm of the mapping from d to y, or for reference tracking on the H norm of r e. For both mappings an acceptable performance is achieved, if the following condition on the sensitivity function S = (I + PC) 1 is met: γw 1 (I + PC) 1 1 (35) 1 This condition is fulfilled, if S(jω) W 1. Consequently, a typical weighting (jω) function W 1 (s) has a shape as shown in figure 18. An integrator in W 1 (s) forces September 19, 211 Raoul Herzog, Jürg Keller 23

24 Advanced Control, An Overview on Robust Control MSE the sensitivity function to be zero at ω =. The parameter γ < 1 is used as an optimization parameter. The larger γ the better the disturbance attenuation. Performance and robust stability can therefore be achieved, if the two conditions are combined to the following condition (mixed sensitivity approach): Additive uncertainty: Multiplicative uncertainty: [ ] γw1 (I + PC) 1 W 2,a C(I + PC) 1. (36) [ γw 1 (I + PC) 1 W 2,m PC(I + PC) 1 ]. (37) Figure 19: Control System for mixed Sensitivity Optimization The block diagram representation of the mixed sensitivity problem is shown in 19. Here it becomes obvious, that the robust stability condition can be viewed as a mapping of signals. A controller can be found by solving the following optimization problem: Additive uncertainty: max γ (Similarly for the multiplicative uncertainty) [ ] γw1 (s)(i + PC) 1 W 2,a (s)c(i + PC) 1 = 1, (38) Figure 2: LFT for H optimization 24 Raoul Herzog, Jürg Keller September 19, 211

25 MSE Advanced Control, An Overview on Robust Control With some matrix algebra, the equation (38) can be transformed into a lower linear fractional transformation F(P, C) as shown in figure 2. Therefore, the following problem has to be solved: Find a stabilizing controller C, for which: F(P,C) := P 11 + P 12 C(I P 22 C) 1 P 21 = 1 (39) Nowadays, various solutions to this problem are known. The two most important are state space methods based on the solution of two Riccati equation and the solution based on linear matrix inequalities (LMI). Special attention has to be payed on additional assumptions, that are imposed for the algorithms. Some of them will be treated in the next chapter. 6 A Glimpse on the H State Space Solution H controller design problems can be formulated in different ways, but finally, the problem has to be brought to a representation as shown in figure 2. Its state space representation is: ẋ = Ax + B 1 w + B 2 u (4) z = C 1 x + D 11 w + D 12 u (41) y = C 2 x + D 21 w + D 22 u (42) The system matrices are usually represented as follows: (43) A. B 1 B 2 G s =. (44) C 1. D 11 D 12 C 2. D 21 D 22 A closed optimal solution for the above general system is not yet published. For the H problem a closed solution was first developped in [2] for the following special case: A. B 1 B 2 [ ] G s = C 1.. (45) I [ ] C 2. I In a design tool the general state space description of equation (44) is transformed into the representation of equation (45) by means of loop shifting [3]. The transformation of D 21 and D 12 into the form with the identity matrix, requires that the two September 19, 211 Raoul Herzog, Jürg Keller 25

26 Advanced Control, An Overview on Robust Control MSE matrices have full rank. This condition is usually violated, when weighting functions are strictly proper, i.e. have a zero D matrix. To circumvent this problem, the weighting D-matrix can usually be set to a small value, without influencing the controller design. The conditions for existence of a solution are: (A,B 2 ) is stabilisable and (C 2,A) is detectable D 12 and D 21 must be of full rank [ ] A jωi B2 has ful column rank for all ω C 1 D 12 [ ] A jωi B1 has ful column rank for all ω C 2 D 21 Another important aspect is that the H problem must have a solution at ω =. An unappropriate weighting at ω = can lead to unsatisfactory controller designs. For a closed loop system with plant as in equation (44) the direct feedthrough term (gain at ω = ) is: D cl = D 11 + D 12 (I D c D 22 ) 1 D c D 21 (46) where D c is the controller D-matrix. The term D 11 is modified with the controller D c -matrix. This can only be done to a limited extent, which is depending on the sizes of D 12 and D 21. Bounds are documented in [3]. 7 Limitation of H Methods In this section some of the limitations of H controller design will be summarized. The H controller design methods offers powerful tools to solve various design problems. Nevertheless H performance measures are not always adequate for the investigated design problem and although robustness and performance measures are combined in the design procedure, it does not really guarantee, that performance is robust with respect to plant uncertainty. Consequently it was proposed to combine H robustness measures with other performance measures, for example measures similar to the LQR-design, the mixed H 2 /H design. Also LMI (see chapter outlook ) offers many possibilities for mixed design problems. H controller design leads to controllers of high order. A high order controller needs more resources for its implementation and is suspect to numerical problems. Therefore it is favorable to have low order controllers. The optimal solution for a system as described by equation ((44)) has the same order as 26 Raoul Herzog, Jürg Keller September 19, 211

27 MSE Advanced Control, An Overview on Robust Control the system itself (size of the A-matrix). The order of the system is the order of the plant to be controlled and the sum of the orders of all the weighting functions, see figure 19. As already mentioned in the section on uncertainty description, the weighting functions have to be of minimal order. The problem of high order controllers can be diminished with order reduction methods. There are several order reduction methods available. These can be applied either on plant before H controller design or after design on the controller itself. The designer should be cautious not to spoil the controller design with a radical order reduction. There are also suboptimal H controller design algorithm which allow a direct design of a reduced order controller, but these are not currently available in the commercially available controller design tools, although a direct low order controller design is doubtless the way to go. As it can be seen from figures 11 the uncertainty description is not tight in the sense, that it incorporates a much larger set of plants than necessary. This leads to a conservative controller design, which might result in very poor closed loop performance. This problem is reduced, when a structured uncertainty description is used in conjuction with µ synthesis (see Chapter Outlook ). 8 Outlook: µ Synthesis and LMI Methods 8.1 Structured Singular Values (SSV) and µ Synthesis The standard H minimizes an H norm between input signal w and output signal z. Often these signals are vector valued (MIMO), as it is already the case for z in the mixed sensitivity approach. In practice we are interested to individual transfer functions between a scalar component w k and a scalar component z l. A large amount of conservativeness is introduced in the design when optimizing an overall H norm. The µ framework allows a more selective optimization related to structured uncertainty. A detailed description is out of the scope of this document. Basically the µ framework introduces a new norm based on structured singular values which is the new quantity to be minimized. The solution procedure is iterative (D K iteration) and involves a sequence of minimizations, first over the controller K (holding the scaling variable D associated with the µ), and then optimizing over the scaling D (holding the controller K fixed). The D K procedure is not guaranteed to converge neither to the global minimum nor to a local minimum of the µ value, but often works well in practice. It has been applied to a number of real world applications with success. These applications include vibration suppression for flexible structures, flight control, chemical process control, and acoustic reverberation suppression in enclosures. 8.2 Linear Matrix Inequalities (LMI) Linear Matrix Inequalities (LMI) have emerged as powerful numerical design tools in areas ranging from robust control (e.g. H ) to system identification. September 19, 211 Raoul Herzog, Jürg Keller 27

28 Advanced Control, An Overview on Robust Control MSE The canonical form of linear matrix inequality is any constraint of the form M(v) = M + v 1 M 1 + v 2 M v n M n < (47) where v = [v 1,...,v n ] is a vector of unknowns called decision variables, and M,...,M n are given symmetric matrices. M(v) < stands for negative definite, i.e. all eigenvalues of M(v) are negative. The scalar decision variables v 1,...,v n are often related to unknown matrices in control problems, e.g. quadratic Lyapunov functions V (x) = x T P x with P >. Here the entries of the symmetric matrix P correspond to decision variables. It is common to express LMI s in matrix form instead of the scalar formulation (47). There are three generic LMI problems: Feasibility problem: Find a solution v to the LMI system M(v) <. The set of all solutions is also called feasibility set. A simple example for an LMI feasibility problem is the determination of stability. The system ẋ = Ax is stable if there exist a Lyapunov function V (x) = x T P x > with a negative derivative V = A T P + PA <. This turns out to be an LMI problem with an unknown matrix P. Another frequent example is a state space system with bounded infinity norm. This turns out to be an LMI problem. It is easy to show that the feasibility set of equation (47) is convex. Convexity has an important consequence: even though the optimization problem has no analytical solution in general, it can be solved numerically with a guaranteed convergence to the solution when one exists. This is in sharp contrast to general nonlinear optimization algorithms which may not converge towards the global optimum. v 2 v 2 v 1 v 1 convex constraint non-convex constraint Figure 21: Convexity Linear objective minimization problem: Minimize the linear function c T v subject to the LMI constraint M(v) <. Generalized eigenvalue minimization problem: Minimize λ subject to M(v) < λ B(v), B(v) >, and C(v) <. Efficient 15 interior point algorithms [4] are available to solve these three generic 15 However the complexity of LMI computations can grow quickly with the problem order. For example, the number of operations required to solve a standard Riccati equation is o(n 3 ) where n is the state dimension. Solving an equivalent LMI Riccati inequality needs o(n 6 ) operations! 28 Raoul Herzog, Jürg Keller September 19, 211

29 MSE Advanced Control, An Overview on Robust Control LMI problems. Many control problems and general design specifications can be formulated in terms of LMI. The main strength of LMI formulations is the ability to combine various convex design constraints or objectives in a numerically tractable manner [5] [6]. A non exhaustive list of problems addressed by LMI techniques include the following: 1. robust pole placement 2. optimal LQG control 3. robust H control 4. multi objective synthesis 5. robust stability of systems with structured parameter uncertainty (µ analysis) It is fair to say the advent of LMI optimization has significantly influenced the direction of research in robust control. A widely accepted technique for numerically solving robust control problems is to reduce them to LMI problems. 9 Conclusion Robust control emerged around 198, and the progress in theory and numerical algorithms during the last 25 years has been enormous. Efficient commercial and public domain software tools are available nowadays. It is possible to successfully use these tools without understanding the deepest details of the underlying theory. However, a minimum of theoretical knowledge is necessary, and as it is the case with any complex scientific software tools 16, a lot of practical experience is needed before being able to solve real world 17 problems. In robust control, specific reasons for this are the following: Modeling is a challenging engineering task, and finding an estimation the modeling uncertainty needs some experience. Additionally, either control specifications are not entirely known in the beginning, i.e. incomplete, or real world specifications may be very complex 18 to such an extent that no direct synthesis procedure exist. No matter which controller synthesis method is used it remains an iterative process which also needs some experience. In robust control, the user often needs to play around with frequency weighting functions in order to understand the performance/robustness trade offs of his specific problem. One of the main achievements in robust control might be the consciousness about the industrial importance of robustness. Generally speaking, system failure is not accepted in our society, and robustness considerations will remain very important. 16 e.g. tools for finite element modeling, or any other complex scientific tool 17 = non academic 18 e.g. mixed time domain and frequency domain specifications, low controller order, etc. September 19, 211 Raoul Herzog, Jürg Keller 29